Abelian groups

Definition

Groups + xy = yx

Examples

• Integers modulo n, (under addition modulo n), which is notated variously as C_n, Z_n, Z/(n). For example, C_4 has the multiplication table

                   | 0 1 2 3 
                  -+--------
                  0| 0 1 2 3
                  1| 1 2 3 0
                  2| 2 3 0 1
                  3| 3 0 1 2
           

• The free abelian group on k generators is isomorphic to Z^k, the direct product of k copies of the integers.

Structure

Every finitely generated abelian group is isomorphic to some

              Z_{q_1} x Z_{q_2} x ... x Z_{q_n} x Z^r
        

where q_1 divides q_2 divides ... divides q_n and r >= 0. r is called the rank of the abelian group, and q_i the invariants. The rank and invariants are uniquely determined by the group. For example, there are exactly six nonisomorphic abelian groups of order 360 = 2^3 . 3^2 . 5, namely

             Z_360
             Z_2 x Z_180
             Z_2 x Z_2 x Z_90
             Z_2 x Z_6 x Z_30
             Z_3 x Z_120
             Z_6 x Z_60
        

Representation

Decision problems

Identity problem: Solvable

Word problem: Solvable

Spectra and growth

Finite spectrum: 1,1,1,2,1,1,1,3,2,1,...

Free spectrum: 1, aleph_null, aleph_null, ...

Growth series: ((1+z)/(1-z))^r for the free abelian group on r generators

History/Importance

References

[Kurosh]

Subsystems

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A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu